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Register a new userPeriod-index and u-invariant questions for function fields over complete discretely valued fields
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Let K be a complete discretely valued field and F the function field of a curve over K. If the characteristic of the residue field k of K is p > 0, then we give a bound for the Brauer p-simension of F in terms of the p-rank of k. If k is a perfect field of characteristic 2, we show that the u-invaraint of F is at most 8.
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Let K be a complete discretely valued field with residue field k. If char(K) = 0, char(k) = 2 and the 2-rank of k is d, we prove that there exists an integer N depending on d such that the u-invariant of any function field in one variable over K is bounded by N. The method of proof is via introducing the notion of uniform boundedness for the p-torsion of the Brauer group of a field and relating the uniform boundedness of the 2-torsion of the Brauer group to finiteness of the u-invariant. We prove that the 2-torsion of the Brauer group of function fields in one variable over K are uniformly bounded.
A topological flux function is introduced to quantify the topology of magnetic braids: non-zero line-tied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a cross-section of the magnetic field, whose integral over the cross-section yields the relative magnetic helicity. Recognising that the topological flux function is an action in the Hamiltonian formulation of the field line equations, a simple formula for its differential is obtained. We use this to prove that the topological flux function uniquely characterises the field line mapping and hence the magnetic topology. A simple example is presented.
A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a classification of finite-dimensional graded-central graded-division algebras over an arbitrary field $mathbb{F}$ can be reduced to the following three classifications, for each finite Galois extension $mathbb{L}$ of $mathbb{F}$: (1) finite-dimensional central division algebras over $mathbb{L}$, up to isomorphism; (2) twisted group algebras of finite groups over $mathbb{L}$, up to graded-isomorphism; (3) $mathbb{F}$-forms of certain graded matrix algebras with coefficients in $Deltaotimes_{mathbb{L}}mathcal{C}$ where $Delta$ is as in (1) and $mathcal{C}$ is as in (2). As an application, we classify, up to graded-isomorphism, the finite-dimensional graded-division algebras over the field of real numbers (or any real closed field) with an abelian grading group. We also discuss group gradings on fields.
We answer two open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two result follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.
Let $E$ be a finite directed graph, and let $I$ be the poset obtained as the antisymmetrization of its set of vertices with respect to a pre-order $le$ that satisfies $vle w$ whenever there exists a directed path from $w$ to $v$. Assuming that $I$ is a tree, we define a poset of fields over $I$ as a family $mathbf K = { K_i :iin I }$ of fields $K_i$ such that $K_isubseteq K_j$ if $jle i$. We define the concepts of a Leavitt path algebra $L_{mathbf K} (E)$ and a regular algebra $Q_{mathbf K}(E)$ over the poset of fields $mathbf K$, and we show that $Q_{mathbf K}(E)$ is a hereditary von Neumann regular ring, and that its monoid $mathcal V (Q_{mathbf K}(E))$ of isomorphism classes of finitely generated projective modules is canonically isomorphic to the graph monoid $M(E)$ of $E$.
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